课程名称︰总体经济理论一
课程性质︰必修(经济硕/博)
课程教师︰毛庆生
开课学院：社会科学院
开课系所︰经济学研究所
考试日期（年月日）︰2013年1月某日
考试时限（分钟）：180分钟
试题 :
Final Examination: Macroeconomic Theory (Fall 2012)
[110 pts.total]
"以下小t,t-1,t+1,t+j...等皆为下标，代表第t,t-1,t+1,t+j期间"
"d0及E0则代表第0期"
1.Lucas Tree Model
Consider the tree model discussed in the lecture. The dividend stream from
the tree is stochastic. Agents in this economy trade a risky share of the
tree and a risk-free one period bond. The representative consumer is posited
to solve the following maximization problem(notations follow my lectures):
∞
max E0[Σ (β^t)*u(ct)], 0<β<1
{ct,zt+1,bt+1} t=0
subject to (ct)+(bt+1)+(qt)*(zt+1)=(1+rt)*(bt)+[(qt)+(dt)]*(zt), ∀t
Equilibrium requires ct=dt, zt=1 and bt=0 for all t.
(1)[5 pts.]
Derive the pricing equations for bond and share. Interpret briefly.
(2)[10 pts.]
Assume dividend evolves according to dt=(dt-1)*(1+μ)*(εt), where
ln(εt) is identically and independently distributed as N(0,σε^2).
The utility function is given by u(c)=c^(1-γ)/(1-γ) if γ≠1 and
u(c)=ln(c) if γ=1. Please derive a closed-form solution for the
equilibrium price-dividend ratio qt/dt and the real interest rate
rt+1. [Hint:E(εt)=exp(σε^2/2). Also ,assume β(1+μ)^(1-γ)<1.]
(3)[10 pts.]
Consider your solution in (2). Suppose the random shock εt rises at
time t. What is the effect of this disturbance on the equilibrium
share price qt and the real interest rate rt+1? I expect to see analysis
of the bond market and the stock market.
(4)[10 pts.]
Condiser your solution in (2). Suppose the growth rate μ rises
unexpectedly at time t. How will the share prices qt and real interest rate
rt+1 respond? Draw expected time path of q and r. Justify your results.
Again, I expected to see analysis of the bond market and the stock market.
(5)[10 pts.]
Assume divdend evolves according to dt=(1+μ)'*εt, where ln(εt) is as
assumed in (2). The utility function is also of equilibium price-diviend
ratio qt/dt.
(6)[10 pts.]
Consider your solution in (5). Suppose the random shock εt rises at time t.
What is the effect of this disturbance on the equilibrium share price qt
and the real interest rate rt+1? Justify your results.
(7)[5 pts.]
Continuing last question. Assume εt rises by 1%. What is the percentage
change in qt? How is your answer related to the parameterγ? Justify your
quantitative assessment.
2.CKR Model
Consider the stochastic CKR model discussed in the lecture. The
representative household is postulated to solve the following problem
(notations are standard):
∞
max E0[Σβ^t*u(ct)]
{ct,kt+1} t=0
subject to ct+[(kt+1)-(1-δ)kt]+Gt=yt=λt*f(kt),
λt follows a stationary random process.
(1)[10 pts.]
Write down the Bellman's equation and derive the equilibrium conditions
of the model.
(2)[10 pts.]
Consider a market economy where identical consumers and firms trade in
a competitive goods market, a competitive labor market and a competitive
risk-free bond market. The representative consumter inelastically supplies
one unit of labor(nt=1 for all t) and solves the following maximization
problem(wt=real wage rate, dt=divident remitted from firms, other variables
are as usual):
∞
max E0[Σ β^t*u(ct)]
{ct,bt+1} t=0
subject to ct+bt+1=wt+dt+(1+rt)*(bt),∀t
Given the marginal utility of consumption u'(ct) determined by consumers,
the represetative firm solves the following maximization problem:
∞
max E0[Σ β^t*u'(ct)*(dt)]
{nt,kt+1} t=0
subject to dt=λt*F(kt,nt)-wt*nt-it,
it=kt+1-(1-δ)kt, δ∈(0,1),∀t
Note that, although labor supply is fixed, firms have to decide how much
labor to hire, The production function is assumed to exhibit CRTS.
Equilibrium requires the goods market, the labor market and the bond market
clear at all dates. Please define a competitive equilibrium for this economy
and show that the equilibrium coincides with the optimal solution in (1).
[Hint: When nt=1, yt=(λt)*F(kt,nt=1)=(λt)*f(kt).]
(3)[5 pts.]
Suppose u(c)=ln(c) and y=(λt)*k^α, α∈(0,1). Calculate the determinisitc
steady state of k, c, y, r and w. Discuss the steady state effect of a
permanent rise in λ.
(4)[15 pts.]
Analyze the effects of a permanent rise in λt. Draw the impulse responses
of capital, consumption, output and the real interest rate. Explain
intuitively your results. I expect to see detail analysis of market
equilibrium from date to date.
(5)[10 pts.]
Suppose the production function is yt=F(kt,nt)+λt, that is, the shock is
additive instead of proportional. Analyze the effects of a permanent rise
in λt and draw the impulse responses of capital, consumption, output and
the real interest rate. Why is the result of this case different form (4)?
Explain.
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